# Why You Might Want to Bootstrap

A few weeks back, Quanta Magazine had an article about attempts to “bootstrap” the laws of physics, starting from simple physical principles and pulling out a full theory “by its own bootstraps”. This kind of work is a cornerstone of my field, a shared philosophy that motivates a lot of what we do. Building on deep older results, people in my field have found that just a few simple principles are enough to pick out specific physical theories.

There are limits to this. These principles pick out broad traits of theories: gravity versus the strong force versus the Higgs boson. As far as we know they don’t separate more closely related forces, like the strong nuclear force and the weak nuclear force. (Originally, the Quanta article accidentally made it sound like we know why there are four fundamental forces: we don’t, and the article’s phrasing was corrected.) More generally, a bootstrap method isn’t going to tell you which principles are the right ones. For any set of principles, you can always ask “why?”

With that in mind, why would you want to bootstrap?

First, it can make your life simpler. Those simple physical principles may be clear at the end, but they aren’t always obvious at the start of a calculation. If you don’t make good use of them, you might find you’re calculating many things that violate those principles, things that in the end all add up to zero. Bootstrapping can let you skip that part of the calculation, and sometimes go straight to the answer.

Second, it can suggest possibilities you hadn’t considered. Sometimes, your simple physical principles don’t select a unique theory. Some of the options will be theories you’ve heard of, but some might be theories that never would have come up, or even theories that are entirely new. Trying to understand the new theories, to see whether they make sense and are useful, can lead to discovering new principles as well.

Finally, even if you don’t know which principles are the right ones, some principles are better than others. If there is an ultimate theory that describes the real world, it can’t be logically inconsistent. That’s a start, but it’s quite a weak requirement. There are principles that aren’t required by logic itself, but that still seem important in making the world “make sense”. Often, we appreciate these principles only after we’ve seen them at work in the real world. The best example I can think of is relativity: while Newtonian mechanics is logically consistent, it requires a preferred reference frame, a fixed notion for which things are moving and which things are still. This seemed reasonable for a long time, but now that we understand relativity the idea of a preferred reference frame seems like it should have been obviously wrong. It introduces something arbitrary into the laws of the universe, a “why is it that way?” question that doesn’t have an answer. That doesn’t mean it’s logically inconsistent, or impossible, but it does make it suspect in a way other ideas aren’t. Part of the hope of these kinds of bootstrap methods is that they uncover principles like that, principles that aren’t mandatory but that are still in some sense “obvious”. Hopefully, enough principles like that really do specify the laws of physics. And if they don’t, we’ll at least have learned how to calculate better.

# Calculating the Hard Way, for Science!

I had a new paper out last week, with Jacob Bourjaily and Matthias Volk. We’re calculating the probability that particles bounce off each other in our favorite toy model, N=4 super Yang-Mills. And this time, we’re doing it the hard way.

The “easy way” we didn’t take is one I have a lot of experience with. Almost as long as I’ve been writing this blog, I’ve been calculating these particle probabilities by “guesswork”: starting with a plausible answer, then honing it down until I can be confident it’s right. This might sound reckless, but it works remarkably well, letting us calculate things we could never have hoped for with other methods. The catch is that “guessing” is much easier when we know what we’re looking for: in particular, it works much better in toy models than in the real world.

Over the last few years, though, I’ve been using a much more “normal” method, one that so far has a better track record in the real world. This method, too, works better than you would expect, and we’ve managed some quite complicated calculations.

So we have an “easy way”, and a “hard way”. Which one is better? Is the hard way actually harder?

To test that, you need to do the same calculation both ways, and see which is easier. You want it to be a fair test: if “guessing” only works in the toy model, then you should do the “hard” version in the toy model as well. And you don’t want to give “guessing” any unfair advantages. In particular, the “guess” method works best when we know a lot about the result we’re looking for: what it’s made of, what symmetries it has. In order to do a fair test, we must use that knowledge to its fullest to improve the “hard way” as well.

We picked an example in the middle: not too easy, and not too hard, a calculation that was done a few years back “the easy way” but not yet done “the hard way”. We plugged in all the modern tricks we could, trying to use as much of what we knew as possible. We trained a grad student: Matthias Volk, who did the lion’s share of the calculation and learned a lot in the process. We worked through the calculation, and did it properly the hard way.

Which method won?

In the end, the hard way was indeed harder…but not by that much! Most of the calculation went quite smoothly, with only a few difficulties at the end. Just five years ago, when the calculation was done “the easy way”, I doubt anyone would have expected the hard way to be viable. But with modern tricks it wasn’t actually that hard.

This is encouraging. It tells us that the “hard way” has potential, that it’s almost good enough to compete at this kind of calculation. It tells us that the “easy way” is still quite powerful. And it reminds us that the more we know, and the more we apply our knowledge, the more we can do.

# QCD Meets Gravity 2019

I’m at UCLA this week for QCD Meets Gravity, a conference about the surprising ways that gravity is “QCD squared”.

When I attended this conference two years ago, the community was branching out into a new direction: using tools from particle physics to understand the gravitational waves observed at LIGO.

At this year’s conference, gravitational waves have grown from a promising new direction to a large fraction of the talks. While there were still the usual talks about quantum field theory and string theory (everything from bootstrap methods to a surprising application of double field theory), gravitational waves have clearly become a major focus of this community.

This was highlighted before the first talk, when Zvi Bern brought up a recent paper by Thibault Damour. Bern and collaborators had recently used particle physics methods to push beyond the state of the art in gravitational wave calculations. Damour, an expert in the older methods, claims that Bern et al’s result is wrong, and in doing so also questions an earlier result by Amati, Ciafaloni, and Veneziano. More than that, Damour argued that the whole approach of using these kinds of particle physics tools for gravitational waves is misguided.

There was a lot of good-natured ribbing of Damour in the rest of the conference, as well as some serious attempts to confront his points. Damour’s argument so far is somewhat indirect, so there is hope that a more direct calculation (which Damour is currently pursuing) will resolve the matter. In the meantime, Julio Parra-Martinez described a reproduction of the older Amati/Ciafaloni/Veneziano result with more Damour-approved techniques, as well as additional indirect arguments that Bern et al got things right.

Before the QCD Meets Gravity community worked on gravitational waves, other groups had already built a strong track record in the area. One encouraging thing about this conference was how much the two communities are talking to each other. Several speakers came from the older community, and there were a lot of references in both groups’ talks to the other group’s work. This, more than even the content of the talks, felt like the strongest sign that something productive is happening here.

Many talks began by trying to motivate these gravitational calculations, usually to address the mysteries of astrophysics. Two talks were more direct, with Ramy Brustein and Pierre Vanhove speculating about new fundamental physics that could be uncovered by these calculations. I’m not the kind of physicist who does this kind of speculation, and I confess both talks struck me as rather strange. Vanhove in particular explicitly rejects the popular criterion of “naturalness”, making me wonder if his work is the kind of thing critics of naturalness have in mind.

# Rooting out the Answer

I have a new paper out today, with Jacob Bourjaily, Andrew McLeod, Matthias Wilhelm, Cristian Vergu and Matthias Volk.

There’s a story I’ve told before on this blog, about a kind of “alphabet” for particle physics predictions. When we try to make a prediction in particle physics, we need to do complicated integrals. Sometimes, these integrals simplify dramatically, in unexpected ways. It turns out we can understand these simplifications by writing the integrals in a sort of “alphabet”, breaking complicated mathematical “periods” into familiar logarithms. If we want to simplify an integral, we can use relations between logarithms like these:

$\log(a b)=\log(a)+\log(b),\quad \log(a^n)=n\log(a)$

to factor our “alphabet” into pieces as simple as possible.

The simpler the alphabet, the more progress you can make. And in the nice toy model theory we’re working with, the alphabets so far have been simple in one key way. Expressed in the right variables, they’re rational. For example, they contain no square roots.

Would that keep going? Would we keep finding rational alphabets? Or might the alphabets, instead, have square roots?

After some searching, we found a clean test case. There was a calculation we could do with just two Feynman diagrams. All we had to do was subtract one from the other. If they still had square roots in their alphabet, we’d have proven that the nice, rational alphabets eventually had to stop.

So we calculated these diagrams, doing the complicated integrals. And we found they did indeed have square roots in their alphabet, in fact many more than expected. They even had square roots of square roots!

You’d think that would be the end of the story. But square roots are trickier than you’d expect.

Remember that to simplify these integrals, we break them up into an alphabet, and factor the alphabet. What happens when we try to do that with an alphabet that has square roots?

Suppose we have letters in our alphabet with $\sqrt{-5}$. Suppose another letter is the number 9. You might want to factor it like this:

$9=3\times 3$

Simple, right? But what if instead you did this:

$9=(2+ \sqrt{-5} )\times(2- \sqrt{-5} )$

Once you allow $\sqrt{-5}$ in the game, you can factor 9 in two different ways. The central assumption, that you can always just factor your alphabet, breaks down. In mathematical terms, you no longer have a unique factorization domain.

Instead, we had to get a lot more mathematically sophisticated, factoring into something called prime ideals. We got that working and started crunching through the square roots in our alphabet. Things simplified beautifully: we started with a result that was ten million terms long, and reduced it to just five thousand. And at the end of the day, after subtracting one integral from the other…

We found no square roots!

After all of our simplifications, all the letters we found were rational. Our nice test case turned out much, much simpler than we expected.

It’s been a long road on this calculation, with a lot of false starts. We were kind of hoping to be the first to find square root letters in these alphabets; instead it looks like another group will beat us to the punch. But we developed a lot of interesting tricks along the way, and we thought it would be good to publish our “null result”. As always in our field, sometimes surprising simplifications are just around the corner.

# Calabi-Yaus in Feynman Diagrams: Harder and Easier Than Expected

I’ve got a new paper up, about the weird geometrical spaces we keep finding in Feynman diagrams.

With Jacob Bourjaily, Andrew McLeod, and Matthias Wilhelm, and most recently Cristian Vergu and Matthias Volk, I’ve been digging up odd mathematics in particle physics calculations. In several calculations, we’ve found that we need a type of space called a Calabi-Yau manifold. These spaces are often studied by string theorists, who hope they represent how “extra” dimensions of space are curled up. String theorists have found an absurdly large number of Calabi-Yau manifolds, so many that some are trying to sift through them with machine learning. We wanted to know if our situation was quite that ridiculous: how many Calabi-Yaus do we really need?

So we started asking around, trying to figure out how to classify our catch of Calabi-Yaus. And mostly, we just got confused.

It turns out there are a lot of different tools out there for understanding Calabi-Yaus, and most of them aren’t all that useful for what we’re doing. We went in circles for a while trying to understand how to desingularize toric varieties, and other things that will sound like gibberish to most of you. In the end, though, we noticed one small thing that made our lives a whole lot simpler.

It turns out that all of the Calabi-Yaus we’ve found are, in some sense, the same. While the details of the physics varies, the overall “space” is the same in each case. It’s a space we kept finding for our “Calabi-Yau bestiary”, but it turns out one of the “traintrack” diagrams we found earlier can be written in the same way. We found another example too, a “wheel” that seems to be the same type of Calabi-Yau.

We also found many examples that we don’t understand. Add another rung to our “traintrack” and we suddenly can’t write it in the same space. (Personally, I’m quite confused about this one.) Add another spoke to our wheel and we confuse ourselves in a different way.

So while our calculation turned out simpler than expected, we don’t think this is the full story. Our Calabi-Yaus might live in “the same space”, but there are also physics-related differences between them, and these we still don’t understand.

At some point, our abstract included the phrase “this paper raises more questions than it answers”. It doesn’t say that now, but it’s still true. We wrote this paper because, after getting very confused, we ended up able to say a few new things that hadn’t been said before. But the questions we raise are if anything more important. We want to inspire new interest in this field, toss out new examples, and get people thinking harder about the geometry of Feynman integrals.

# Congratulations to Simon Caron-Huot and Pedro Vieira for the New Horizons Prize!

The 2020 Breakthrough Prizes were announced last week, awards in physics, mathematics, and life sciences. The physics prize was awarded to the Event Horizon Telescope, with the $3 million award to be split among the 347 members of the collaboration. The Breakthrough Prize Foundation also announced this year’s New Horizons prizes, six smaller awards of$100,000 each to younger researchers in physics and math. One of those awards went to two people I know, Simon Caron-Huot and Pedro Vieira. Extremely specialized as I am, I hope no-one minds if I ignore all the other awards and talk about them.

The award for Caron-Huot and Vieira is “For profound contributions to the understanding of quantum field theory.” Indeed, both Simon and Pedro have built their reputations as explorers of quantum field theories, the kind of theories we use in particle physics. Both have found surprising behavior in these theories, where a theory people thought they understood did something quite unexpected. Both also developed new calculation methods, using these theories to compute things that were thought to be out of reach. But this is all rather vague, so let me be a bit more specific about each of them:

Simon Caron-Huot is known for his penetrating and mysterious insight. He has the ability to take a problem and think about it in a totally original way, coming up with a solution that no-one else could have thought of. When I first worked with him, he took a calculation that the rest of us would have taken a month to do and did it by himself in a week. His insight seems to come in part from familiarity with the physics literature, forgotten papers from the 60’s and 70’s that turn out surprisingly useful today. Largely, though, his insight is his own, an inimitable style that few can anticipate. His interests are broad, from exotic toy models to well-tested theories that describe the real world, covering a wide range of methods and approaches. Physicists tend to describe each other in terms of standard “virtues”: depth and breadth, knowledge and originality. Simon somehow seems to embody all of them.

Pedro Vieira is mostly known for his work with integrable theories. These are theories where if one knows the right trick one can “solve” the theory exactly, rather than using the approximations that physicists often rely on. Pedro was a mentor to me when I was a postdoc at the Perimeter Institute, and one thing he taught me was to always expect more. When calculating with computer code I would wait hours for a result, while Pedro would ask “why should it take hours?”, and if we couldn’t propose a reason would insist we find a quicker way. This attitude paid off in his research, where he has used integrable theories to calculate things others would have thought out of reach. His Pentagon Operator Product Expansion, or “POPE”, uses these tricks to calculate probabilities that particles collide, and more recently he pushed further to other calculations with a hexagon-based approach (which one might call the “HOPE”). Now he’s working on “bootstrapping” up complicated theories from simple physical principles, once again asking “why should this be hard?”

# At Aspen

I’m at the Aspen Center for Physics this week, for a workshop on Scattering Amplitudes and the Conformal Bootstrap.

Aspen is part of a long and illustrious tradition of physics conference sites located next to ski resorts. It’s ten years younger than its closest European counterpart Les Houches School of Physics, but if anything its traditions are stricter: all blackboard talks, and a minimum two-week visit. Instead of the summer schools of Les Houches, Aspen’s goal is to inspire collaboration: to get physicists to spend time working and hiking around each other until inspiration strikes.

This workshop is a meeting between two communities: people who study the Conformal Bootstrap (nice popular description here) and my own field of Scattering Amplitudes. The Conformal Boostrap is one of our closest sister-fields, so there may be a lot of potential for collaboration. This week’s talks have been amplitudes-focused, I’m looking forward to the talks next week that will highlight connections between the two fields.